On the support of the solution of a system of quasi-variational inequalities

Abstract

The functions ur(x), u2(x) characterize the optimal cost of a stochastic control problem, whose decision variables are switching points. The results on existence, uniqueness and regularity of (l), (2) are recalled in Section 1, where we also describe the stochastic control problem which corresponds to it. We are interested here in obtaining regions which belong to the set of points where ur = u2 + k, or u2 = ur + k, , denoted by S, , S, respectively, and called the supports of the solution of (l), (2). Th’ IS p ro bl em is the analog of the problem on the support of solutions of elliptic variational inequalities (v.i.) considered by Brezis [6]. For the q.v.i. arising in the impulse control problem, studied by Bensoussan and Lions [3], the analogous question of support has been considered recently by Bensoussan, Brezis and Friedman [5]. The general methodology that we apply relies on the results on support for vi. However, we also use here the special characteristic of (l), (2) in order to obtain sharp estimates on the support, which is the main objective of the paper. We notice that if g, = g, = g then ur - u2 = w is a solution of the v.i.